Furthermore ([1]}), for any \(T>0\) ,
\(\Big (\int _0^T\Big \Vert \int _0^t e^{i(t-t^{\prime }) \partial _x^2} f(\cdot ,t^{\prime })dt^{\prime }\Big \Vert ^{q_1}_{L_x^{p_1}}dt\Big )^{1/{q_1}}\lesssim \Big (\int _0^T\Vert f(\cdot ,t^{\prime })\Vert ^{q^{\prime }_0}_{L_x^{p^{\prime }_0}}dt\Big )^{1/{q^{\prime }_0}},\) 
